Bases and interbasis transformations for the $SU(2)$ monopole

The variables in the Schrödinger equation for the bound "charge–$SU(2)$-monopole" system are separated in hyperspherical, parabolic, and spheroidal coordinates in the space $\mathbb R^5$. It is shown that the expansion coefficients of the parabolic basis with respect to the hyperspherical basis can be expressed in terms of the Clebsch–Gordon coefficients of the group $SU(2)$. Three-term recurrence relations are derived for the expansion coefficients of the spheroidal basis with respect to the hyperspherical and parabolic bases.